On the generalized Lie structure of associative algebras

Journal of Algebra, 1985

This paper studies commutative bialgebras (graded or ungraded, over a field k of arbitrary characteristic) and their relationship to Lie coalgebras. In particular, if A is a commutative bialgebra then the vector space Q.4 = A +/(A +)' (where A + is the augmentation ideal Ker E of A) inherits from A the structure of a Lie coalgebra. One of our aims is to consider what can be said about A if the Lie coalgebra QA is given. This involves a new treatment of Lie coalgebras K and their (universal) coenveloping coalgebras (denoted here UcK); the latter have the structure of commutative Hopf algebras. (Lie coalgebras have previously been considered in [An, An2, Mi, Nil; in the finite-dimensional case, Lie coalgebras and Lie algebras are just dual spaces of each other.) At prime characteristic we make an additional assumption on A, namely, that A be a r-bialgebra (also known as divided power bialgebra or bialgebra with divided powers). A r-bialgebra, by definition, is a bialgebra A such that, first, the underlying algebra of A has the structure of a r-algebra, that is, is commutative and has a sequence {yiJiaO of operators with properties like those which the operators .xbxi/i! have at characteristic 0 (the precise definition is given in (3.1) below), and second, the coalgebra operations are compatible with the operators yi. (At characteristic 0, r-bialgebra = commutative bialgebra.) r-bialgebras A have been studied [An, An2, GL, SC, Sj] in the case in which A = Ciao Ai is graded and connected (that is, A, = k), and have applications to algebraic topology (homology of Eilenberg-MacLane spaces) and commutative algebra (Tor of a local ring). The principal result on connected graded f-bialgebras, due to Andre [An2], says (in our notation) that a connected graded r-bialgebra A is isomorphic to the coenveloping coalgebra UcK of

ARABIAN JOURNAL FOR SCIENCE AND ENGINEERING

This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative algebra and the classification of these structures in low dimensions.

2018

The aim of this paper is to introduce n-ary BiHom-algebras, generalizing BiHom-algebras. We introduce an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibniz algebra and study various type of n-ary BiHom-Lie algebras and BiHom-associative algebras. We show that n-ary BiHom-Lie-Leibniz algebra can be represented by BiHom-Lie-Leibniz algebra through fundamental objets. Moreover, we provide some key constructions and study n-ary BiHom-Lie algebras induced by (n-1)-ary BiHom-Lie algebras.

Cornell University - arXiv, 2022

In this article, we mainly study the products of commutator ideals of Lie-admissible algebras such as Novikov algebras, bicommutative algebras, and assosymmetric algebras. More precisely, we first study the properties of the lower central chains for Novikov algebras and bicommutative algebras. Then we show that for every Lie nilpotent Novikov algebra or Lie nilpotent bicommutative algebra A, the ideal of A generated by the set {ab − ba | a, b ∈ A} is nilpotent. Finally, we study properties of the lower central chains for assosymmetric algebras, study the products of commutator ideals of assosymmetric algebras and show that the products of commutator ideals have a similar property as that for associative algebras.

Israel Journal of Mathematics, 2012

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Dergr(A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Dergr(A)).

Journal of Algebra, 1980

Following V. N. Latylev [9] a Lie algebra ~5 over a c~mrn~~at~ve ring k is called an SPI-algebra if there exists an associative over k and a k-linear embedding i: L + A such that for all x: y i([x, y]) = i(x) i(y)-d(y) i(x)* other words, if we consider A as a Lie algebra under [x, y] = my-yx: en L; is isomorphic to a Lie subalgebra of A. 1.4. ur results depend on standard theorems from the theory of associative PI-algebras. Most of them can be foun in Procesi [II].

Bulletin of the Belgian Mathematical Society, Simon Stevin

In this note we provide a general method to study the endomorphism rings of H-comodule algebras over subalgebras. With this method we may derive the endomorphism rings of crossed products and some duality theorems for graded rings.

arXiv: Rings and Algebras, 2018

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and coproduct. We here give a classification of connected, cocommutative Com-PreLie bialgebras over a field of characteristic zero: we obtain a main family of symmetric algebras on a space V of any dimension, and another family available only if V is one-dimensional. We also explore the case of Com-PreLie bialgebras over a group algebra and over a tensor product of a group algebra and of a symmetric algebra.

Publicacions Matemàtiques, 2008

In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra Q is an algebra of quotients of a Lie algebra L in terms of the associative algebras generated by the adjoint operators of L and Q respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients.

Journal of Algebra, 2015

Let F be a field of characteristic not 2 . An associative F -algebra R gives rise to the commutator Lie algebra R (−) = (R, [a, b] = ab − ba). If the algebra R is equipped with an involution * : R → R then the space of the skew-symmetric elements K = {a ∈ R | a * = −a} is a Lie subalgebra of R (−) . In this paper we find sufficient conditions for the Lie algebras [R, R] and [K, K] to be finitely generated.

2021

Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are derived. The bimodules, matched pairs and Manin triple of a nearly associative algebras are derived and their equivalence with nearly associative bialgebras is proved. Basic definitions and properties of nearly Hom-associative algebras are described. Related bimodules and matched pairs are given, and associated identities are established.

Communications in Algebra, 2013

A finite-dimensional Lie algebra L over a field F of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in [10].

Advances in Mathematics, 2008

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra (g,A)(F), which is the Lie subalgebra of F⊗A generated by F⊗g. In many examples A is the universal enveloping algebra